Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 22 (2026), 049, 34 pages      arXiv:2412.04579      https://doi.org/10.3842/SIGMA.2026.049

Solvable Families of Random Block Tridiagonal Matrices

Brian Rider a and Benedek Valkó b
a) Department of Mathematics, Temple University, Philadelphia, PA, USA
b) Department of Mathematics, University of Wisconsin - Madison, Madison, WI, USA

Received November 05, 2025, in final form April 23, 2026; Published online May 14, 2026

Abstract
We introduce two families of random tridiagonal block matrices for which the joint eigenvalue distributions can be computed explicitly. These distributions are novel within random matrix theory, and exhibit interactions among eigenvalue coordinates beyond the typical mean-field log-gas type. Leveraging the matrix models, we go on to describe the point process limits at the edges of the spectrum in two ways: through certain random differential operators, and also in terms of coupled systems of diffusions. Along the way we establish several algebraic identities involving sums of Vandermonde determinant products.

Key words: random matrices; beta-ensembles; eigenvalue distribution.

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